Finite Element Approximation of Rotating MHD-Boussinesq Flows with Nonlinear Power-Law Viscosity
Abstract
We propose and analyse a fully discrete finite element approximation of the three-dimensional incompressible rotating MHD-Boussinesq system with nonlinear power-law (p-Laplace) viscosity. Conforming elements are used for the velocity and temperature, H(curl)-conforming Nédélec elements for the magnetic induction, and a semi-implicit backward Euler scheme in time. The discretisation preserves the principal energy structure: the Lorentz and induction terms cancel exactly, while the Coriolis force remains energy neutral. This yields well-posedness and mesh-independent energy stability. The nonlinear power-law diffusion creates two main analytical difficulties. Since a uniform time-derivative estimate for the discrete velocity is unavailable, strong compactness is obtained through a time-translate argument and Simon's compactness criterion. The weak limit of the nonlinear diffusion is then identified by a Minty argument based on a discrete energy limsup inequality. Compactness and passage to the limit hold throughout the shear-thickening regime, while identification of a genuine weak solution is obtained in the Newtonian case and, in three dimensions, above an explicit threshold dictated by the Lorentz force rather than by convection. We further establish a priori error estimates without any mesh-time coupling condition. Three-dimensional numerical experiments confirm the stability and convergence behaviour predicted by the analysis.
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BibTeX
@misc{panda2026finite,
title = {{Finite Element Approximation of Rotating MHD-Boussinesq Flows with Nonlinear Power-Law Viscosity}},
author = {Akash Ashirbad Panda and Evana Islam Sarkar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31091},
eprint = {2610.31091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31091},
}
RIS
TY - UNPB
TI - Finite Element Approximation of Rotating MHD-Boussinesq Flows with Nonlinear Power-Law Viscosity
AU - Akash Ashirbad Panda
AU - Evana Islam Sarkar
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31091
UR - https://arxiv.org/abs/2610.31091
AB - We propose and analyse a fully discrete finite element approximation of the three-dimensional incompressible rotating MHD-Boussinesq system with nonlinear power-law (p-Laplace) viscosity. Conforming elements are used for the velocity and temperature, H(curl)-conforming Nédélec elements for the magnetic induction, and a semi-implicit backward Euler scheme in time. The discretisation preserves the principal energy structure: the Lorentz and induction terms cancel exactly, while the Coriolis force remains energy neutral. This yields well-posedness and mesh-independent energy stability. The nonlinear power-law diffusion creates two main analytical difficulties. Since a uniform time-derivative estimate for the discrete velocity is unavailable, strong compactness is obtained through a time-translate argument and Simon's compactness criterion. The weak limit of the nonlinear diffusion is then identified by a Minty argument based on a discrete energy limsup inequality. Compactness and passage to the limit hold throughout the shear-thickening regime, while identification of a genuine weak solution is obtained in the Newtonian case and, in three dimensions, above an explicit threshold dictated by the Lorentz force rather than by convection. We further establish a priori error estimates without any mesh-time coupling condition. Three-dimensional numerical experiments confirm the stability and convergence behaviour predicted by the analysis.
ER -